1

Multi-scale Circular Conebeam InteriorTomography using Bedrosian Identity: Verification

with Real DataYo Seob Han1, Minji Lee1, John Paul Ward2, Michael Unser3, Seungryoung Cho4 and Jong Chul Ye1,∗

Abstract—Circular trajectory is quite often used in conebeamCT (CBCT) such as C-arm CT, dental CT and so on. However, ifthe cone angle is wide, the FDK algorithm suffers from conebeamartifacts. Moreover, it exhibits severe truncation artifacts ifthe detector is truncated in transverse-ways. To mitigate theseartifacts, we propose a reconstruction method that consists oftwo steps: multi-scale interior tomography using 1D TV in bothhorizontal and vertical virtual chord lines, which is followed byspectral blending in Fourier domain. For spectral blending, wedevelop a Fourier domain analysis technique to identify the miss-ing frequency regions and design a bow tie window for weighting.Experimental results with a real head phantom confirm thatthe proposed method significantly improves the reconstructionquality and reduces the computational time significantly.

Index Terms—Conebeam artifact, Interior tomography,Bedrosian theorem, Multiscale decomposition, Spectral Blending.

I. INTRODUCTION

C IRCULAR CBCT trajectory has been widely used inpractice since the trajectory can be easily implemented

in hardware compared to other geometries such as helical [1]or saddle [2] trajectories. In the circular trajectory, the FDKalgorithm is the de facto standard, but it suffers from theconebeam artifacts as the cone angle increases. These artifactsbecome more severe when only part of detector is used forimaging the region-of-interior (ROI) to reduce the radiationdose.

Specifically, interior tomography approaches reduce the x-ray dose by preventing x-ray illumination outside of the ROI.However, due to the detector truncation, the conventionalfiltered back projection type algorithm cannot be used. Toaddress this problem, the authors in [3] showed that if the ob-ject is essentially piecewise constant, then ROI can be solveduniquely and stably via the total variation (TV) minimization.However, this methods requires 2D or 3D total variation min-imization and iterative applications of forward and backwardprojections, which is computationally very expensive.

1Bio Imaging & Signal Processing Lab., Dept. of Bio and BrainEngineering, Korea Advanced Institute of Science & Technology(KAIST), 291 Daehak-ro, Yuseong-gu, Daejon, Korea. (minjilee@kaist.ac.kr,hanyoseob@kaist.ac.kr, jong.ye@kaist.ac.kr).

2Department of Mathematics, University of Central Florida, Orlando, FL32815. (john.ward@ucf.edu).

3Biomedical Imaging Group, Ecole Polytechnique Federale de Lausanne(EPFL), CH-1015, Lausanne, Switzerland. (michael.unser@epfl.ch).

4Medical Imaging & Radiotherapeutic Lab., Dept. of Bio and BrainEngineering, Korea Advanced Institute of Science & Technology (KAIST),291 Daehak-ro, Yuseong-gu, Daejon, Korea. (scho@kaist.ac.kr).

In interior tomography problems from circular trajectory,two types of artifacts reside: one from truncated detectors andthe other from missing frequency regions. To address the firstartifact, our group recently proposed the multi-scale interiortomography algorithm using 1D TV [4]. Then, to reduce thesecond type artifacts, we also extended the Fourier blendingidea proposed for half-scan FBP algorithm [5] to our multi-scale interior tomography approach, and provided a novelFourier domain two-way weighting scheme [4]. Unlike theoriginal Fourier blending idea in [5], our method is basedon rigorous analysis of Fourier components. The resultingalgorithm is computationally so efficient that it can be easilyused in a clinical environment.

The main goal of this paper is, therefore, to demostrate theeffectiveness of this algorithm using real data. For this, we firstreview the recent theory of multi-scale circular conebeam inte-rior tomography using Bedrosian identity and spectal blending[4] and provide experimental results using real data.

II. THOERY

A. Conebeam artifact problem

1) Fourier analysis of DBP on virtual chord lines: In the3-D CBCT problem, let the variables θ denote a vector on theunit sphere S2 ∈ R3. Then, the x-ray transform is formallydefine as

Df (a,θ) =

∫ ∞0

f(a + tθ)dt , (1)

where f corresponds to the linear attenuation coefficients andθ ∈ S2 denotes the x-ray photon propagation direction anda ∈ R3 refers the x-ray source location in a actual sourcetrajectory a(λ), λ ∈ [λmin, λmax].

For the given source trajectory a(λ), we now define thedifferentiated backprojection (DBP):

g(x) =

∫ λ+

λ−dλ

1

‖x− a(λ)‖∂

∂νDf (a(ν),θ)|ν=λ (2)

where [λ−, λ+] ⊂ [λmin, λmax] denotes the appropriate inter-vals from the source segments between λmin and λmax, and1/‖x− a(λ)‖ denotes the distance weighting. The chord lineon the DBP data can be representated as a Hilbert transformrelationship [1], [6], [7]. Unfortunately, in CBCT, the Hilberttransform relationship is established only on the actual chordline. As our novel contribution, we generalized the Hilberttransform relationship from the actual chord line to a virtualchord line [4].

2

Actual trajectory

Virtual trajectory

Actual chord line

Virtual chord line

Actual source

Virtual source

Fig. 1. Source trajectories and virtual chord lines. The dark circle is an actualtrajectory and the dark line is an actual chord line. The set of gray-dot circleare virtual trajectory and the dark-dot line is a virtual chord line.

The virtual coordinate system can be explained by threeunit vectors which are the z-axis ez , the chord line direction(i.e., filtering direction) e, and their perpendicular axis e⊥.Using the coordinate system, a new coordinate (x′, y′, z) isrepresentated from a primary Cartesian coordinate such that

x = x′e + y′e⊥ + zez . (3)

At the new coordinate, the corresponding spatial frequencyis defined by (wx′ , wy′ , wz). Then, we have the followinggerenalized Hilbert transform relationship.

Theorem II.1. [4] Let the source trajectory a(λ), λ ∈[λ−, λ+] have no discontinuities. Suppose, furthermore, posi-tion x is on the viritual chord line that connects the two virtualsource positions av(λ

−) and av(λ+), and has the coordinate

values (x′, y′, z) on the virtual chord line coordinate system(3). Then, the differentiated backprojection data in (2) can berepresented as

g(x) =1

2π

∫ ∞−∞

dωx′ φ(ωx′ , y′, z)j sgn(ωx′)e

jωx′x′

(4)

where

φ(ωx′ , y′, z) :=

1

(2π)2

∫dωx′

∫ωz∈{(ωx′ ,ωy′ ,ωz)/∈N (z)}

dωz

f(ωx′ , ωy′ , ωz)ej(y′ωy′+zωz) ,

and the missing frequency set N (z) on z is given by

N (z) =

{(ωx′ , ωy′ , ωz) | −A ≤

ωx′

zωz≤ B

}, (5)

where

A =1

x′ +√R2 − (y′)2

, B =1√

R2 − (y′)2 − x′.

which trivially becomes an empty set when z = 0.

2) Missing frequency regions: Thanks to Theorem II.1, wecan identify the missing frequency region. Fig. 2 shows variousmissing frequency regions depends on the filtering direction.If all of the chord lines head for the horizontal direction (blue

(a)

(b) (c)

(d) (e)

Fig. 2. The missing frequency region. (a) The top view of a point x =(x, y, z) and an source trajectory. The blue line indicates the horizontaldirectional virtual chord line between av(λ

−1 ) and av(λ

+1 ) and the red

line denotes the vertical directional virtual chord lines between av(λ−2 ) and

av(λ+2 ). (b)(c) The missing frequency region for horizontal and vertical

directions, respectively. (d) (e) The missing frequency region for the diagonaldirectional virtual chord lines that correspond to the yellow and green lines,respectively.

line at Fig. 2(a)), then the missing frequency region for a givenz can be written as{

(ωx, ωy, ωz) | −zωz

x+√R2 − y2

≤ ωx ≤zωz√

R2 − y2 − x

}, where zωz ≤ 0; on the contrary, for the vertical direction(red line at Fig. 2(b)), the missing frequency region can beexplained such that{

(ωx, ωy, ωz) | −zωz

y +√R2 − x2

≤ ωy ≤zωz√

R2 − x2 − y

}.

The missing frequency region is illustrated in Fig. 2(b)(c)for x � z, y � z, respectively. In addition, Fig. 2(d)(e) rep-resents the missing frequency regions related to the diagonalfiltering directions.

B. Multi-scale Interior Tomography using Bedrosian Identity

In interior tomography problems, an available DBP data ona virtual chord line is truncated within FOV xπ ∈ (e1, e2). Ac-cordingly, when Hilbert transform is applied to the truncateddata, we can expect truncation artifacts since there exists anull space:

HhN (xπ) = 0, where xπ ∈ (e1, e2). (6)

Here

hN (xπ) =1

π

∫R\(e1,e2)

dx′πψπ(x′π)

x′π − xπ, (7)

for some function ψπ(x). Thus, we are interested in imposing1D TV to correct any data hN (xπ) belong to the null space.

By decomposing interior tomgoraphy problems based on the1D TV formulations, the computational cost is much lowerthan it is for high-dimensional TV [3], [8], [9]. Moreover, wecan significantly reduce the computational complexity basedon the Bedrosian identity of Hilbert transform.

3

(a) (b) (c) (d)Fig. 3. The common missing frequency region according to the filteringdirections. (a), (b), and (c) indicate the missing frequency region accordingto the number of missing frequency regions with different filtering directions.The dark region denotes a common missing frequency region, respectively.(d) Overlapped illustration of the common missing frequency regions.

Theorem II.2. (see [10]) Let w, f ∈ L2(R) be low-pass andhigh-pass signals such that the Fourier transform of w(x)vanishes for |ω| > ω0, with ω0 > 0, and the Fourier transformof f(x) vanishes for |ω| < ω0. Then, we have

H{w(x)f(x)} = w(x)H{f(x)} . (8)

Accordingly, the interior tomography problem is formulatedas

minfL‖fL‖TV (L;E) subject to g = H(fL + fH) . (9)

where fL and fH denote the low and high freqeuncy compo-nent of a object f , respectively. By solving Eq. (9), the lowfreqeuncy component fL(x) is reconstructed under TV con-straint. Moreover, it is possible to reconstruct low-resolutionimage in the down-sampled domain, so the computational bun-den is significantly reduced. From the reconstruncted signalfL(x), the residual signal gH(x) is extracted by substractingHfL(x) from g(x). If gH(x) does not overlap with the spectralband-width of the truncation window, then Theorem II.2 issatisfied, so, fH(x) is directly calculated by

fH(x) =−H{w(x)gH(x)}

w(x), x ∈ E. (10)

The multiscale decomposition method is summarized inFig. 4.

C. Conebeam artifact reduction using spectral blending

However, the recovered signal is not the exact solutiondue to the missing frequency regions. Indeed, the best signalwe can expect is that the signals with missing frequencycomponents as shown in Fig. 2. Therefore, to minimize thecone beam artifacts that are recovered from the interior to-mography algorithm, we need spectral blending that optimallycombines the interior tomography reconstruction in multiplefiltering directions so that the resulting missing frequencycomponents can be reduced to those of Fig. 3. Moreover, based

Fig. 4. Flowchart of the proposed multiscale reconstruction approach forinterior tomography problem.

(a) (b) (c)Fig. 5. ROI tomography reconstruction results using (a) BPF, (b) the pro-posed multiscale reconstruction method along horizontal (φ(x)) and vertical(φ⊥(x)) filter directions, and (c) their the spectral blending results from thetwo filtering directions.

on Fig. 3, the size of the missing frequency regions from twoorthogonal filtering directions is still comparable to that ofmultiple filtering directions. Since additional filtering directionrequires additional applications of iterative interior topographyalgorithm, the computational complexity increases; so thispaper just utilities the two filtering directions along x- andy- axis and apply the optimal spectral blending.

More specifically, as shown in the Fig. 3, it is possibleto minimize the missing frequency region by blendingappropriate frequency components of the reconstruction fromthe two orthogonal filtering directions. The shape of missingfrequency region is described as Fig. 2(b) owing to thehorizontal filtering direction, then it can be minimized byapplying the row-wise bow tie window like Fig. 5(c) bluewindow. On the contrary, the shape is described as Fig. 2(c)owing to the vertical filtering direction, then it can be alsominimized by applying the column-wise bow tie window likeFig. 5(c) red window. Finally, by blending both the weightedspectrums, the missing frequency region can be minimizedin the 2D Fourier domain in each slice. Therefore, we apply2D Fourier transform for each z-slice and use the spectralblending, which significantly reduces the computationalburden. The concept of spectral blending simply is illustratedin Fig. 5. The multiscale decomposition interior tomographyalgorithm is used for both horizontal and vertical directions,respectively. Then, using spectral weighting with a bow-tiewindow, they are blended into one image.

III. RESULTS

The reconstruction domain resolution of the real headphantom was 512× 512× 512 voxles with voxel size 0.589×0.589 × 0.684 mm3, and the phantom size is (−150, 150) ×(−150, 150) × (−175, 175) mm3. The resolution of the de-tector is 1024 × 250 array matrix with detector pitch of0.4× 0.4 mm2 and the number of views is 720. The distancefrom source to rotation axis is 1700 mm, and the distancefrom source to detector is 2250 mm. The radius of FOV wasabout 36 mm. Since a transverse-ways offset is applied as 5.0pitch, when the projection data was acquired, the ROI of thereconstructed head phantom is biased toward the right-bottomside.

4

912.0/0.403

370.2/0.164

83.7/0.037

917.7/0.339

385.2/0.142

103.1/0.038

1773.5/1.979

497.2/0.555

97.2/0.109

full-beam BPF FDK BPF TV Profile

89.2/0.039

96.1/0.036

120.7/0.135

Proposed

−20 0 20−1000

0

1000

2000

3000

4000

y position (mm)

Housefield Unit

PhantomFDKBPFTVProposed

−20 0 20−1000

0

1000

2000

3000

4000

x position (mm)

Housefield Unit

PhantomFDKBPFTVProposed

−20 0 20−1000

0

1000

2000

3000

4000

z position (mm)

Housefield Unit

PhantomFDKBPFTVProposed

Fig. 6. The reconstruction results of real head phantom. From top to bottom, the each row denotes the cross section for the transection at z = −11.3 mm(i.e. off-mid-plane), the coronal at y = −3.8 mm, and the sagittal at x = −0.9 mm. The profiles corresponds to the CT numbers along the white lines ofthe reconstructed results. The profiles of full-beam BPF, FDK, BPF, TV and proposed method are represented by gray-dot, green, blue, and red, respectively.

In Fig. 6, reconstruction results for a real head phantom areshown in the region (−36, 36)× (−36, 36)× (−36, 36) mm3.Although the proposed method is truncated as about 70%,the results in Fig. 6 are very similar to that of the full-beam result. Unlike the TV reconstruction, the noise texturescould be recovered by the proposed method, and confirmedthat the algorithm does not result in unnatural smoothing.This is because TV is only applied at the low frequencyreconstruction, and the high frequency reconstruction is donean using analytic formula.

When the computation time is compared between the pro-posed and conventional iterative (TV) method, a performanceof reconstruction is 2.2 slice/sec for the proposed method,however, in the conventional case, the performance is 0.0667slice/sec. This shows that the proposed method is 34.3 timesaccelerated over the conventional.

IV. CONCLUSION

In this paper, we provided a novel analysis of the missingfrequency region using Fourier domain analysis, which isdistinct from the Radon domain approach in [5]. Basedon the analysis, an optimal spectral blending scheme thatweights the reconstruction from two orthogonal filteringdirections was proposed. We further demonstrated that theDBP data from the circular cone beam data has a similarHilbert transform relationship on virtual chord lines, butthe content is different owing to the missing frequencyregions. Accordingly, our algorithm consisted of two stepreconstruction procedure: first, the multiscale decompositioninterior tomography algorithm using 1D TV penalty on thevirtual chord lines, which is followed by spectral blending ofa two reconstructions from horizontal and vertical filtering

directions. The proposed method provided the reconstructionresult with significantly reduced conebeam and missingfrequency artifacts. Furthermore, all the processing were donein 1D virtual chord lines, which significantly reduces thecomputational complexity.

This work was supported by Korea Science and EngineeringFoundation by Grant NRF- 2014R1A2A1A11052491.

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